Abrupt transition in the structural formation of interconnected networks

Abstract

Our world is linked by a complex mesh of networks through which information, people and goods flow. These networks are interdependent on each other, and present structural and dynamical features 1-6 different from those observed in isolated networks 7-9 . Although examples of such dissimilar properties are becoming more abundant-such as in diffusion, robustness and competition-it is not yet clear where these differences are rooted. Here we show that the process of building independent networks into an interconnected network of networks undergoes a structurally sharp transition as the interconnections are formed. Depending on the relative importance of inter-and intra-layer connections, we find that the entire interdependent system can be tuned between two regimes: in one regime, the various layers are structurally decoupled and they act as independent entities; in the other regime, network layers are indistinguishable and the whole system behaves as a singlelevel network. We analytically show that the transition between the two regimes is discontinuous even for finite-size networks. Thus, any real-world interconnected system is potentially at risk of abrupt changes in its structure, which may manifest new dynamical properties.Interacting, interdependent or multiplex networks are different ways of naming the same class of complex systems where networks are not considered as isolated entities but interacting with each other. In multiplex, the nodes at each network are instances of the same entity; thus, the networks are representing simply different categorical relationships between entities, and usually categories are represented by layers. Interdependent networks is a more general framework where nodes can be different at each network.Many, if not all, real networks are coupled with other real networks. Examples can be found in several domains: social networks (for example, Facebook, Twitter and so on) are coupled because they share the same actors 10 ; multimodal transportation networks are composed of different layers (for example, bus, subway and so on) that share the same locations 11 ; the functioning of communication and power grid systems depends one on the other 1 . So far, all phenomena that have been studied on interdependent networks, including percolation 1,3 , epidemics 4 and linear dynamical systems 5 , have provided results that differ much from those valid in the case of isolated complex networks. Sometimes the difference is radical: for example, whereas isolated scale-free networks are robust against failures of their nodes or edges 12 , scale-free interdependent networks are instead very fragile 1,3 .Given such observations, two fundamentally important theoretical questions are in order: Why do dynamical and critical phenomena running on interdependent network models differ

Structural reducibility of multilayer networks

“…Similarly, in biological systems basic constituents such as proteins can have physical, co-localization, genetic or many other types of interactions. Recently, it has been shown that retaining such multi-dimensional information 7 and modelling the structure of interdependent and multilayer systems respectively through interdependent 8 and multilayer networks [9][10][11][12] reveals new nontrivial structural properties [13][14][15][16][17][18][19][20] and relevant emergent physical phenomena [21][22][23][24][25][26][27] .…”

An Ensemble Perspective on Multi-layer Networks

“…Note that the change between λ 2 and either λ 1 2 or λ 2 2 being maximal is related to the transitions pointed out in Ref. [3,16]. This behavior is shown in Fig.…”

Characterization of multiple topological scales in multiplex networks through supra-Laplacian eigengaps

“…For continuity, at p * ,μ 3 = mp must hold, since mp is always an eigenvalue of the supra-Laplacian. p = p * is the point at which the structural transition described in [7,14] occurs, as already noted by Darabi Sahneh et al [15]. Each eigenvalue up toμ n will follow the same pattern, following the lineμ i = mp and departing from it to approach its boundμ (a) i when it hits the next eigenvalueμ i+1 = mp (see Fig.…”

“…So far, theoretical studies have pointed out that this is indeed the case [4][5][6]. Moreover, as recently shown [7,8], multiplex networks might show different structural phases. Namely, under some conditions, the multiplex system might behave as one interconnected system, while in other conditions, the layers can become effectively disconnected and behave as if they were isolated [9].…”

“…Finally, it is informative to look at the quantum entropy of M. S q (M) shows a clear peak after p * and before p (see the inset of Fig. 4), i.e., in the region after the transition observed in [7,14] and before the one we have introduced here. In fact, by studying the sign of the derivative of S q , it can be proven that the quantum entropy must have a peak before p .…”